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 TITLE : Unit 01: Place Value of Whole Numbers and Decimals SUGGESTED DURATION : 10 days

Unit Overview

Introduction
This unit bundles student expectations that address representing the value of whole numbers and decimals, interpreting each place-value position using multiples of 10, and comparing and ordering these numbers. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 3, students composed and decomposed numbers up to 100,000, identified base-10 relationships through the hundred thousands place, and compared and ordered these numbers. They also represented fractions with models and number lines. Previous grade levels used the decimal point in money only.

During this Unit
Students extend their understanding of patterns in place value to represent the value of the digits in whole numbers through one billion and decimals to the hundredths using expanded notation and numerals. Students further generalize the value of each place-value position as 10 times the value of the place to its right and as one-tenth of the value of the place to its left. Students compare and order whole numbers to one billion and represent comparisons using symbols. Ordering three or more numbers may include situations involving quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.) and may involve the location of the numbers on a number line. Students represent decimals, including tenths and hundredths, using concrete and visual models (e.g., number lines, decimal disks, decimal grids, base-10 blocks) and money. Students also determine the corresponding decimal to the tenths or hundredths place of a specified point on a number line. Comparing and ordering decimals to the hundredths is accomplished using concrete and visual models (e.g., place-value charts, number lines, decimal disks, decimal grids, base-10 blocks, money, etc.).

Other considerations: Reference the Mathematics COVID-19 Implementation Tool Grade 4

After this Unit
In Unit 02, students will further examine the role of 10 in the base-10 place value system when rounding numbers to the hundred thousands place and when rounding to the nearest 10, 100, or 1,000 to estimate solutions in problems involving whole numbers. Additionally, students will apply place value understandings to add and subtract whole numbers and decimals to the hundredths place. In Grade 5, students will represent the value of the digit in decimals through the thousandths place, compare and order decimals through the thousandths, and use place-value knowledge to round decimals to tenths and hundredths.

In Grade 4, the representation of whole numbers and decimals is identified as STAAR Readiness Standard 4.2B, while mathematical relationships found in the base-10 place value system is identified as STAAR Supporting Standard 4.2A. Comparing whole numbers is addressed by STAAR Supporting Standard 4.2C. Representing decimals with models as well as the comparison of decimals using models are identified as STAAR Supporting Standards 4.2E and 4.2F. Locating decimals on a number line and representing decimals as distances from zero on a number line are identified as STAAR Supporting Standards 4.2H and 4.3G. All of these standards are included in the Grade 4 STAAR Reporting Category 1: Numerical Representations and Relationships and the Grade 4 Texas Response to Curriculum Focal Points (TxRCFP): Understanding decimals and addition and subtraction of decimals. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, B1, B2; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1. A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to Van de Walle & Lovin (2006). “two important ideas typically developed for three-digit numbers should be carefully extended to larger numbers. First, the grouping of ten idea should be generalized. That is, 10 in any position makes a single thing (group) in the next position and vice versa. Second, the oral and written patterns for numbers in three digits are duplicated in a clever way for every three digits to the left” (p. 47). The further state, “It is important for students to realize that the system does have a logical structure, is not totally arbitrary, and can be understood” (p. 49). The National Mathematics Advisory Panel (2008) notes, “By the end of Grade 4, students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals” (p. 20).

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3 – 5. Boston, MA: Pearson Education, Inc.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place. How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• The base-10 place value system, based on 10 digits, allows for communicating very large and very small numbers efficiently (whole numbers through 1,000,000,000).
• The base-10 place value system is based on multiples of 10 where each place represents a relationship of 10 times the value of the place to its right and one-tenth of the value of the place to its left.
• How can any number be formed using only the digits 0 – 9?
• What patterns and relationships are found in the base-10 place value system?
• How does the place value change when …
• moving to the left
• moving to the right
… across the place value positions in a number?
• How are the periods (billions period; millions period; thousands period; hundreds period), the patterns within each period (hundreds place; tens place; ones place), and the comma(s) used to read and write whole numbers?
• What is the purpose of the digit zero in a number and when does the digit zero affect the value of a number?
• A digit’s position within the base-10 place value system determines the value of the number.
• How is the value of a digit within a number determined?
• How does the position of the digits determine the value of a number?
• The ability to recognize and represent numbers in various forms develops the understanding of equivalence and allows for working flexibly with numbers in order to communicate and reason about the value of the number (whole numbers through 1,000,000,000).
• What are some ways a number can be represented?
• What is the relationship between the base-10 place value system language and the way numbers are represented in …
• standard form?
• word form?
• expanded notation?
• Why can a number vary in representation but the value of the number stay the same?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a number using …
• numerals
• expanded notation
… improve understanding and communicating about the value of a number and the equivalence of the representations?
• Quantities are compared and ordered to determine magnitude of number and equality or inequality relations (whole numbers through 1,000,000,000).
• Why is it important to identify the unit or attribute being described by numbers before comparing or ordering the numbers?
• How can …
• place value
• numeric representations
… aid in comparing and/or ordering numbers?
• How can the comparison of two numbers be described and represented?
• How are quantifying descriptors used to determine the order of a set of numbers?
• Number
• Base-10 Place Value System
• Compare and Order
• Comparative language
• Comparison symbols
• Composition and Decomposition of Numbers
• Number
• Counting (natural) numbers
• Whole numbers
• Number Recognition
• Sequence
• Hierarchical inclusion
• Magnitude
• Unitizing
• Number Representations
• Standard form
• Word form
• Expanded form
• Expanded notation
• Relationships
• Numerical
• Equivalence
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place. How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• The base-10 place value system, based on 10 digits, allows for communicating very large and very small numbers efficiently (whole numbers through 1,000,000,000; decimals through hundredths).
• The base-10 place value system is based on multiples of 10 where each place represents a relationship of 10 times the value of the place to its right and one-tenth of the value of the place to its left.
• How can any number be formed using only the digits 0 – 9?
• What patterns and relationships are found in the base-10 place value system?
• How does the place value change when …
• moving to the left
• moving to the right
… across the place value positions in a number?
• How are the periods (billions period; millions period; thousands period; hundreds period), the patterns within each period (hundreds place; tens place; ones place), and the comma(s) used to read and write whole numbers?
• What is the purpose of the digit zero in a number and when does the digit zero affect the value of a number?
• A digit’s position within the base-10 place value system determines the value of the number.
• How is the value of a digit within a number determined?
• How does the position of the digits determine the value of a number?
• Decimal numbers are used in the base-10 place value system to represent a quantity that may include part of a whole.
• What is the purpose of the decimal point in a number?
• What is the purpose of the digit zero in a number and when does the digit zero affect the value of a number?
• What is the relationship between the digits to the left of the decimal point and the digits to the right of the decimal point?
• What patterns exist within the decimal units of a number (tenths; hundredths)?
• The ability to recognize and represent numbers in various forms develops the understanding of equivalence and allows for working flexibly with numbers in order to communicate and reason about the value of the number (whole numbers through 1,000,000,000; decimals through hundredths).
• What are some ways a number can be represented?
• What is the relationship between the base-10 place value system language and the way numbers are represented in …
• standard form?
• word form?
• expanded notation?
• Why can a number vary in representation but the value of the number stay the same?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a number using …
• numerals
• expanded notation
• concrete models
• pictorial models
… improve understanding and communicating about the value of a number and the equivalence of the representations?
• Quantities are compared and ordered to determine magnitude of number and equality or inequality relations (whole numbers through 1,000,000,000; decimals through hundredths).
• Why is it important to identify the unit or attribute being described by numbers before comparing or ordering the numbers?
• How can …
• place value
• numeric representations
• concrete representations
• pictorial representations
… aid in comparing and/or ordering numbers?
• How can the comparison of two numbers be described and represented?
• How are quantifying descriptors used to determine the order of a set of numbers?
• Number
• Base-10 Place Value System
• Compare and Order
• Comparative language
• Comparison symbols
• Composition and Decomposition of Numbers
• Number
• Counting (natural) numbers
• Whole numbers
• Decimals
• Number Recognition
• Sequence
• Hierarchical inclusion
• Magnitude
• Unitizing
• Number Representations
• Standard form
• Word form
• Expanded form
• Expanded notation
• Relationships
• Numerical
• Equivalence
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may not understand that the pattern of following the 0 – 9 sequence and then increasing the digit in the next place goes on to infinity.
• Some students may think that representing the value of the digits in whole numbers using expanded notation can only be written in one way rather than in multiple ways.
• Some students may rely too strongly on money as a decimal model and believe that the decimal point is merely a symbol to separate dollars and cents rather than a symbol to indicate the unit to its immediate left.
• Some students may misinterpret the pattern to the right of the decimal point and think that the word tenths indicates two digits because a certain number of tens requires a symbol that is two digits.
• Some students may use whole number ideas to compare decimals, leading to the belief that a number such as 0.89 is greater than 0.9 or that 0.9 is less than 0.10.

Underdeveloped Concepts:

• Some students may not understand the key role of place value in ordering numbers and may only think about leading digits.

Unit Vocabulary

• Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Digit – any numeral from 0 – 9
• Expanded form – the representation of a number as a sum of place values (e.g., 985,156,789 as 900,000,000 + 80,000,000 + 5,000,000 + 100,000 + 50,000 + 6,000 + 700 + 80 + 9; 985,156,789.78 as 900,000,000 + 80,000,000 + 5,000,000 + 100,000 + 50,000 + 6,000 + 700 + 80 + 9 + 0.7 + 0.08)
• Expanded notation – the representation of a number as a sum of place values where each term is shown as a digit(s) times its place value (e.g., 985,156,789 as (9 × 100,000,000) + (8 × 10,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (6 × 1,000) + (7 × 100) + (8 × 10) + (9 × 1); 985,156,789.78 as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) + 6(1,000) + 7(100) + 8(10) + 9(1) + 7(0.1) + 8(0.01))
• Numeral – a symbol used to name a number
• Order numbers – to arrange a set of numbers based on their numerical value
• Period – a three-digit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
• Standard form – the representation of a number using digits (e.g., 985,156,789.78)
• Strip diagram – a linear model used to illustrate number relationships
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Word form – the representation of a number using written words (e.g., 985,156,789 as nine hundred eighty-five million, one hundred fifty-six thousand, seven hundred eighty-nine; 985,156,789.78 as nine hundred eighty-five million, one hundred fifty-six thousand, seven hundred eighty-nine and seventy-eight hundredths)

Related Vocabulary:

 Ascending Decrease Descending Equality Greater than Hundredth Increase Inequality Interval Leading zero Less than Magnitude Multiple of 10 Number line One-tenth Open number line Tenth Tick marks Trailing zero
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 4 Mathematics TEKS

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
4.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
4.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
4.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
4.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
4.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
4.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
4.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
4.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
4.2 Number and operations. The student applies mathematical process standards to represent, compare, and order whole numbers and decimals and understand relationships related to place value. The student is expected to:
4.2A Interpret the value of each place-value position as 10 times the position to the right and as one-tenth of the value of the place to its left.
Supporting Standard

Interpret

THE VALUE OF EACH PLACE-VALUE POSITION AS 10 TIMES THE POSITION TO THE RIGHT AND AS ONE-TENTH OF THE VALUE OF THE PLACE TO ITS LEFT

Including, but not limited to:

• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
• One billions place
• Hundred millions place
• Ten millions place
• One millions place
• Hundred thousands place
• Ten thousands place
• One thousands place
• Hundreds place
• Tens place
• Ones place
• Tenths place
• Hundredths place
• Base-10 place value system
• A number system using ten digits 0 – 9
• Relationships between places are based on multiples of 10.
• Moving left across the places, the values are 10 times the position to the right.
• Moving right across the places, the values are one-tenth the value of the place to the left.
• Place value relationships and relationships between the values of digits involving whole numbers through (less than or equal to) 1,000,000,000 and decimals to the hundredths (greater than or equal to 0.01)
• Place value relationships are based on multiples of ten whereas relationships between the values of digits may or may not be based on multiples of ten.

Note(s):

• Grade 3 described the mathematical relationships found in the base-10 place value system through the hundred thousands place.
• Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
• Various mathematical process standards will be applied to this student expectation as appropriate .
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
4.2B Represent the value of the digit in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals.

Represent

THE VALUE OF THE DIGIT IN WHOLE NUMBERS THROUGH 1,000,000,000 AND DECIMALS TO THE HUNDREDTHS USING EXPANDED NOTATION AND NUMERALS

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Decimals (great than or equal to 0.01)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Numeral – a symbol used to name a number
• Digit – any numeral from 0 – 9
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
• One billions place
• Hundred millions place
• Ten millions place
• One millions place
• Hundred thousands place
• Ten thousands place
• One thousands place
• Hundreds place
• Tens place
• Ones place
• Tenths place
• Hundredths place
• Base-10 place value system
• A number system using ten digits 0 – 9
• Relationships between places are based on multiples of 10.
• Moving left across the places, the values are 10 times the position to the right.
• Multiplying a number by 10 increases the place value of each digit.
• Moving right across the places, the values are one-tenth the value of the place to the left.
• Dividing a number by 10 decreases the place value of each digit.
• The magnitude (relative size) of whole number places through the billions place
• The magnitude of one billion
• 1,000,000,000 can be represented as 10 hundred millions.
• 1,000,000,000 can be represented as 100 ten millions.
• 1,000,000,000 can be represented as 1,000 one millions.
• The magnitude (relative size) of decimal places through the hundredths
• The magnitude of one-tenth
• 0.1 can be represented as 1 tenth.
• 0.1 can be represented as 10 hundredths.
• The magnitude of one-hundredth
• 0.01 can be represented as 1 hundredth.
• Standard form – the representation of a number using digits (e.g., 985,156,789.78)
• Period – a three-digit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
• Billions period is composed of the one billions place, ten billions place, and hundred billions place.
• Millions period is composed of the one millions place, ten millions place, and hundred millions place.
• Thousands period is composed of the one thousands place, ten thousands place, and hundred thousands place.
• Units period is composed of the ones place, tens place, and hundreds place.
• The word “billion” after the numerical value of the billions period is stated when read.
• A comma between the billions period and the millions period is recorded when written but not stated when read.
• The word “million” after the numerical value of the millions period is stated when read.
• A comma between the millions period and the thousands period is recorded when written but not stated when read.
• The word “thousand” after the numerical value of the thousands period is stated when read.
• A comma between the thousands period and the units period is recorded when written but not stated when read.
• The word “unit” after the numerical value of the units period is not stated when read.
• The word “hundred” in each period is stated when read.
• The words “ten” and “one” in each period are not stated when read.
• The tens place digit and ones place digit in each period are stated as a two-digit number when read.
• The whole part of a decimal number is recorded to the left of the decimal point when written and stated as a whole number.
• The decimal point is recorded to separate the whole part of a decimal number from the fractional part of a decimal number when written and is stated as “and” when read.
• The fractional part of a decimal number is recorded to the right of the decimal point when written.
• The fractional part of a decimal number is stated as a whole number with the label of the smallest decimal place value when read (e.g., 0.5 is read as 5 tenths; 0.25 is read as 25 hundredths; etc.).
• The “-ths” ending denotes the fractional part of a decimal number.
• Zeros are used as place holders between digits of a number as needed, whole part and fractional part, to maintain the value of each digit (e.g., 400.05).
• Leading zeros in a decimal number are not commonly used in standard form, but are not incorrect and do not change the value of the decimal number (e.g., 0,037,564,215.55 equals 37,564,215.55).
• Trailing zeros after a fractional part of a decimal number may or may not be used and do not change the value of the decimal number (e.g., 400.50 equals 400.5).
• Word form – the representation of a number using written words (e.g., 985,156,789.78 as nine hundred eighty-five million, one hundred fifty-six thousand, seven hundred eighty-nine and seventy-eight hundredths)
• The word “billion” after the numerical value of the billions period is stated when read and recorded when written.
• A comma between the billions period and the millions period is not stated when read but is recorded when written.
• The word “million” after the numerical value of the millions period is stated when read and recorded when written.
• A comma between the millions period and the thousands period is not stated when read but is recorded when written.
• The word “thousand” after the numerical value of the thousands period is stated when read and recorded when written.
• A comma between the thousands period and the units period is not stated when read but is recorded when written.
• The word “unit” after the numerical value of the units period is not stated when read and not recorded when written.
• The word “hundred” in each period is stated when read and recorded when written.
• The words “ten” and “one” in each period are not stated when read and not recorded when written.
• The tens place digit and ones place digit in each period are stated as a two-digit number when read and recorded using a hyphen, where appropriate, when written (e.g., twenty-three, thirteen, etc.).
• The whole part of a decimal number is recorded the same as a whole number with all appropriate unit labels prior to recording the fractional part of a decimal number.
• The decimal point is recorded as the word “and” to separate the whole part of a decimal number from the fractional part of a decimal number when written and is stated as “and” when read.
• The fractional part of a decimal number followed by the label of the smallest decimal place value is recorded when written and stated when read.
• The “-ths” ending denotes the fractional part of a decimal number.
• The zeros in a number are not stated when read and are not recorded when written (e.g., 854,091,005.26 in standard form is read and written as eight hundred fifty-four million, ninety-one thousand, five and twenty-six hundredths in word form).
• Place Value forms
• Expanded form –  the representation of a number as a sum of place values (e.g., 985,156,789.78 as 900,000,000 + 80,000,000 + 5,000,000 + 100,000 + 50,000 + 6,000 + 700 + 80 + 9 + 0.7 + 0.08)
• Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 905,150,089.08 as 900,000,000 + 0 + 5,000,000 + 100,000 + 50,000 + 0 + 0 + 80 + 9 + 0.0 + 0.08 or as 900,000,000 + 5,000,000 + 100,000 + 50,000 + 80 + 9 + 0.08).
• Expanded form is written following the order of place value.
• The sum of place values written in random order is an expression but not expanded form.
• Expanded notation – the representation of a number as a sum of place values where each term is shown as a digit(s) times its place value (e.g., 985,156,789.78 as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) + 6(1,000) + 7(100) + 8(10) + 9 + 7(0.1) + 8(0.01) or 985,156,789.78 as 9(100,000,000) + 8(10,000,000) + 5(1,000,000) + 1(100,000) + 5(10,000) + 6(1,000) + 7(100) + 8(10) + 9 + 7() + 8())
• Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 905,150,089.08 as (9 × 100,000,000) + (0 × 10,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (0 × 1,000) + (0 × 100) + (8 × 10) + (9 × 1) + (0 × 0.1) + (8 × 0.01) or as (9 × 100,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (8 × 10) + (9 × 1) + (8 × 0.01) or e.g., 905,150,089.08 as (9 × 100,000,000) + (0 × 10,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (0 × 1,000) + (0 × 100) + (8 × 10) + (9 × 1) + (0 × ) + (8 × ) or as (9 × 100,000,000) + (5 × 1,000,000) + (1 × 100,000) + (5 × 10,000) + (8 × 10) + (9 × 1) + (8 × )).
• Expanded notation is written following the order of place value.
• Expanded notation to represent the value of a digit(s) within a number
• Multiple representations of various forms of a number
• Equivalent relationships between place value of decimals through the hundredths (e.g., 0.2 is equivalent to 20 hundredths).

Note(s):

• Grade 3 composed and decomposed numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate.
• Grade 4 introduces the millions and billions period.
• Grade 4 introduces representing the value of a decimal to the hundredths using expanded notation and numerals.
• Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
4.2C Compare and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >, <, or =.
Supporting Standard

Compare, Order

WHOLE NUMBERS TO 1,000,000,000

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Place value – the value of a digit as determined by its location in a number, such as ones, tens, hundreds, one thousands, ten thousands, etc.
• Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
• Relative magnitude of a number describes the size of a number and its relationship to another number.
• Compare two numbers using place value charts.
• Compare digits in the same place value positions beginning with the greatest value.
• If these digits are the same, continue to the next smallest place until the digits are different.
• Numbers that have common digits but are not equal in value (different place values)
• Numbers that have a different number of digits
• Compare two numbers using a number line.
• Number lines (horizontal/vertical)
• Proportionally scaled number lines (pre-determined intervals with at least two labeled numbers)
• Open number lines (no marked intervals)
• Order numbers – to arrange a set of numbers based on their numerical value
• A set of numbers can be compared in pairs in the process of ordering numbers.
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Order a set of numbers on a number line.
• Order a set of numbers on an open number line.
• Quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)

Represent

COMPARISONS OF WHOLE NUMBERS TO 1,000,000,000 USING THE SYMBOLS >, <, OR =

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Comparative language and comparison symbols
• Inequality words and symbols
• Greater than (>)
• Less than (<)
• Equality words and symbol
• Equal to (=)

Note(s):

• Grade 3 compared and ordered whole numbers up to 100,000 and represented comparisons using the symbols >, <, or =.
• Grade 5 will compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
4.2E Represent decimals, including tenths and hundredths, using concrete and visual models and money.
Supporting Standard

Represent

DECIMALS, INCLUDING TENTHS AND HUNDREDTHS, USING CONCRETE AND VISUAL MODELS AND MONEY

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Decimals (less than or greater than one to the tenths and hundredths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Various concrete and visual models
• Number lines, decimal disks, decimal grids, base-10 blocks, money, etc.

Note(s):

• Previous grade levels used the decimal point in money only.
• Grade 4 introduces representing decimals, including tenths and hundredths, using concrete and visual models and money.
• Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
4.2F Compare and order decimals using concrete and visual models to the hundredths.
Supporting Standard

Compare, Order

DECIMALS USING CONCRETE AND VISUAL MODELS TO THE HUNDREDTHS

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Decimals (less than or greater than one to the tenths and hundredths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
• Comparative language and comparison symbols
• Inequality words and symbols
• Greater than (>)
• Less than (<)
• Equality words and symbol
• Equal to (=)
• Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
• Relative magnitude of a number describes the size of a number and its relationship to another number.
• Compare two decimals using place value charts.
• Compare digits in the same place value position beginning with the greatest place value.
• If these digits are the same, continue to the next smallest place until the digits are different.
• Numbers that have common digits but are not equal in value (different place values)
• Numbers that have a different number of digits
• Compare two decimals with various concrete and visual models.
• Number lines, decimal disks, decimal grids, base-10 blocks, money, etc.
• Order numbers – to arrange a set of numbers based on their numerical value
• Order three or more decimals with various concrete and visual models.
• Quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)
• Number lines, decimal disks, decimal grids, base-10 blocks, money, etc.

Note(s):

• Grade 4 introduces comparing and ordering decimals using concrete and visual models to the hundredths.
• Grade 5 will compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
4.2H Determine the corresponding decimal to the tenths or hundredths place of a specified point on a number line.
Supporting Standard

Determine

THE CORRESPONDING DECIMAL TO THE TENTHS OR HUNDREDTHS PLACE OF A SPECIFIED POINT ON A NUMBER LINE

Including, but not limited to:

• Whole numbers (0 – 1,000,000,000)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Decimals (less than or greater than one to the tenths and hundredths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• All decimals, to the tenths or hundredths place, can be located as a specified point on a number line.
• Characteristics of a number line
• A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
• A minimum of two positions/numbers should be labeled.
• Numbers on a number line represent the distance from zero.
• The distance between the tick marks is counted rather than the tick marks themselves.
• The placement of the labeled positions/numbers on a number line determines the scale of the number line.
• Intervals between position/numbers are proportional.
• When reasoning on a number line, the position of zero may or may not be placed.
• When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoom-in” on the relevant section of the number line.
• Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Characteristics of an open number line
• An open number line begins as a line with no intervals (or tick marks) and no positions/numbers labeled.
• Numbers/positions are placed on the empty number line only as they are needed.
• When reasoning on an open number line, the position of zero is often not placed.
• When working with larger numbers, an open number line without the constraint of distance from zero allows the ability to “zoom-in” on the relevant section of the number line.
• The placement of the first two numbers on an open number line determines the scale of the number line.
• Once the scale of the number line has been established by the placement of the first two numbers, intervals between additional numbers placed are approximately proportional.
• The differences between numbers are approximated by the distance between the positions on the number line.
• Open number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Landmark (or anchor) numbers may be placed on the open number line to help locate other numbers.
• Purpose of open number line
• Open number lines allow for the consideration of the magnitude of numbers and the place-value relationships among numbers when locating a given whole number
• Number lines representing values less than one to the tenths place
• Number lines representing values greater than one to the tenths place
• Number lines representing values less than one to the hundredths place
• Number lines representing values greater than one to the hundredths place
• Number lines representing values between tick marks
• Relationship between tenths and hundredths

Note(s):

• Grade 3 represented a number on a number line as being between two consecutive multiples of 10; 100; 1,000; or 10,000 and use words to describe relative size of numbers in order to round whole numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
4.3 Number and operations. The student applies mathematical process standards to represent and generate fractions to solve problems. The student is expected to:
4.3G

Represent fractions and decimals to the tenths or hundredths as distances from zero on a number line.

Supporting Standard

Represent

DECIMALS TO THE TENTHS OR HUNDREDTHS AS DISTANCES FROM ZERO ON A NUMBER LINE

Including, but not limited to:

• Decimals (less than or greater than one to the tenths and hundredths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Characteristics of a number line
• A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
• A minimum of two positions/numbers should be labeled.
• Numbers on a number line represent the distance from zero.
• The distance between the tick marks is counted rather than the tick marks themselves.
• The placement of the labeled positions/numbers on a number line determines the scale of the number line.
• Intervals between position/numbers are proportional.
• When reasoning on a number line, the position of zero may or may not be placed.
• When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoom-in” on the relevant section of the number line.
• Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Decimals to the tenths or hundredths as distances from zero on a number line
• Relationship between a decimal represented using a strip diagram to a decimal represented on a number line
• Strip diagram – a linear model used to illustrate number relationships
• Decimals as distances from zero on a number line greater than 1
• Point on a number line read as the number of whole units from zero and the decimal amount of the next whole unit
• Number line beginning with a number other than zero
• Distance from zero to first marked increment is assumed even when not visible on the number line.
• Relationship between decimals as distances from zero on a number line to decimal measurements as distances from zero on a metric ruler, meter stick, or measuring tape
• Measuring a specific length using a starting point other than zero on a metric ruler, meter stick, or measuring tape
• Distance from zero to first marked increment not counted
• Length determined by number of whole units and the fractional amount of the next whole unit

Note(s):

• Grade 3 represented fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines.
• Grade 3 determined the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line.
• Grade 5 will represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
• Grade 6 will identify a number, its opposite, and its absolute value.
• Grade 6 will locate, compare, and order integers and rational numbers using a number line.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding decimals and addition and subtraction of decimals
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.